Energetic mid-IR femtosecond pulse generation by self-defocusing soliton-induced dispersive waves in a bulk quadratic nonlinear crystal

Generating energetic femtosecond mid-IR pulses is crucial for ultrafast spectroscopy, and currently relies on parametric processes that, while efficient, are also complex. Here we experimentally show a simple alternative that uses a single pump wavelength without any pump synchronization and without critical phase-matching requirements. Pumping a bulk quadratic nonlinear crystal (unpoled LiNbO$_3$ cut for noncritical phase-mismatched interaction) with sub-mJ near-IR 50-fs pulses, tunable and broadband ($\sim 1,000$ cm$^{-1}$) mid-IR pulses around $3.0~\mu\rm m$ are generated with excellent spatio-temporal pulse quality, having up to 10.5 $\mu$J energy (6.3% conversion). The mid-IR pulses are dispersive waves phase-matched to near-IR self-defocusing solitons created by the induced self-defocusing cascaded nonlinearity. This process is filament-free and the input pulse energy can therefore be scaled arbitrarily by using large-aperture crystals. The technique can readily be implemented with other crystals and laser wavelengths, and can therefore potentially replace current ultrafast frequency-conversion processes to the mid-IR.


Introduction
Intense ultrashort mid-IR (MIR) pulses with microjoule pulse energy are highly important for the study of molecular vibrations [1,2], in particular in the 2,800-4,000 cm −1 region (λ = 2.5 − 3.6 μm) as it contains the stretching modes of C-H, N-H and O-H bonds. The commonly used methods for generating high-energy ultrashort MIR pulses are based on nonlinear optical parametric processes. One of the most popular methods is optical parametric amplification (OPA) [3], and an extra difference-frequency generation (DFG) stage is often integrated to further extend the output wavelength range [4]. These wavelength down-conversion techniques enable the generation of tunable MIR pulses in a broad spectral range and have good conversion efficiencies. The downside is that the processes require critical phase-matching condition and synchronized pump wavelengths, which complicates the setup for MIR generation. Four-wave mixing in χ (3) media has also been reported for intense MIR pulse generation [5][6][7], which also involve synchronized pump wavelengths, and typically bear rather low conversion efficiencies.
Here we conduct an experiment employing a simpler technique: a bulk quadratic nonlinear crystal is pumped with a single near-IR (NIR) femtosecond pulse, and if the pump pulse is intense enough to excite a NIR soliton, it is possible to efficiently transfer some of the soliton energy into an energetic MIR femtosecond pulse. This MIR pulse is generated as a so-called dispersive wave (DW), and the wavelength of the generated MIR radiation is determined by a phase-matching condition between the soliton and the DW. The scheme relies on the excitation of a self-defocusing soliton, which here is done in a standard bulk quadratic χ (2) nonlinear crystal, cut for strongly phase mismatched second-harmonic generation (SHG) and pumped in the normal group-velocity dispersion (GVD) regime. Due to the self-defocusing nature of the soliton the DW will automatically be emitted in the long-wavelength side of the soliton in the anomalous GVD regime, which for the crystal we use (lithium niobate, LN) is in the MIR.
Soliton-induced DWs (also denoted "resonant radiation" or "Cherenkov radiation") have been intensively investigated for generating waves at new frequencies, e.g. in fiber supercontinuum generation [8][9][10]. They were also used for wavelength conversion, especially in the shortwavelength side of the soliton (normal dispersion regime) [11][12][13]. Inside specially designed photonic crystal fibers with small cores and double zero-dispersion wavelengths, fiber-based DWs located to the red side of the soliton were also reported [14][15][16], and with a NIR pump, the longest obtained emission wavelength is about 2.2 μm [16]. One of the major limitations of fiber-based DW generation is the limited pulse energy, especially for DW generation on the red side, which requires fibers with double zero-dispersion wavelengths. Using bulk materials obviously allows to use a much higher pulse energy, but much less effort has been made for this research direction. In bulk, the self-focusing Kerr nonlinearity tends to generate filamentation, and bulk DWs have mainly been observed in connection with conical emission or the characteristic spatio-temporal spectra of the self-focusing filamentation [17][18][19][20]. Another issue is that in bulk self-focusing Kerr media the DW is always located on the blue side of the soliton, such as the UV DWs recently observed [21], and can therefore not be used for downconversion to the MIR.
Using instead a self-defocusing nonlinearity, the soliton formation will be filament free and the DW will be located on the red side of the soliton [22,23]. Such a self-defocusing effect can be generated by phase-mismatched (cascaded) quadratic nonlinearities, e.g. SHG [25]. Under strong phase mismatch, the pump experiences a Kerr-like nonlinearity characterized by the cascaded nonlinear refractive index n I 2,casc ∝ −d 2 eff /Δk, which is negative when the phase-mismatch parameter is positive, Δk = k 2 − 2k 1 > 0. Here d eff is the effective quadratic nonlinearity and k 1 and k 2 are the wavenumbers of the pump and second harmonic (SH), respectively.
We will here investigate an LN crystal, which is attractive because its anomalous dispersion regime lies in the MIR, and this is therefore also the regime where DW emission is predicted [24]. In order to observe such a DW, a NIR self-defocusing soliton must first be excited. The first experimental realization of soliton self-compression and supercontinuum generation in a bulk LN crystal cut for noncritical interaction was demonstrated by some of us [26]. Self-defocusing solitons have also been observed experimentally using bulk crystals cut for critical SHG, namely in periodically poled LN [27,28], and in beta-barium borate (β -BaB 2 O 4 ) [23,[29][30][31]. However, none of these experiments could observe a DW and the closest cases were [23], where a linear NIR wave and a soliton were identified and by calculating the four-wave mixing interaction (a non-degenerate version of the DW phase-matching condition) a DW around 2.0 μm was predicted, and [26] where the NIR soliton was predicted to give DW radiation beyond 3.0 μm. In both of these cases the long-wavelength contents of the spectra could not be measured to confirm this. Supercontinuum generation has also been observed from cascaded SHG in LN waveguides, both in periodically-poled [32,33] and unpoled waveguides [34]. Although a mid-IR peak around 2.8 μm was found in [32], it was later explained by cascaded difference-frequency generation of the soliton and the SH [35], and [34] could not observe DWs as the waveguides did not provide sufficient guiding in the long-wavelength regime.
The goals of the proof-of-principle experiment we here conduct are (a) to show that with a single input pulse it is possible to get μJ-level MIR femtosecond radiation through DW formation in a short piece of nonlinear crystal and (b) to clearly observe and characterize the MIR DW from a self-defocusing cascaded quadratic nonlinearity for the first time. Due to the simple pumping scheme and filament-free nature of the interaction this setup is very scalable in energy. The MIR peak wavelength is tunable by changing the pump wavelength, and the protocol can be generalized to other nonlinear crystals and pumping wavelengths.

Experimental setup
The experimental setup for generating the MIR DW is identical to the one reported in [26]. It is rather simple, consisting of a NIR pump laser source (here a 1 kHz commercial OPA system), which is loosely collimated and projected to the quadratic nonlinear crystal through the telescope consisting of two curved silver reflectors. The beam spot size on the crystal was 0.6 mm full-width-at-half-max (FWHM), which is large enough to avoid beam divergence in the short crystal, and the large spot size is also intended to keep diffraction effects to a minimum. The pulse durations of the NIR pumps were around 50 fs and nearly transform limited. A neutral density filter wheel was used in front of the telescope to adjust the pump intensity. For spectral characterization, an InGaAs CCD-based spectrometer is used for the wavelength below 2300 nm. A grating monochromator and HgCdTe (MCT) detector connected with a box-car integrator was used to record the MIR spectrum.
Just as in [26] we used a 10-mm-long, 5% MgO-doped congruent LN (LiNbO 3 , 10× 10 mm 2 aperture, Altechna). The pump (frequency ω 1 ) and the SH (frequency ω 2 = 2ω 1 ) are both epolarized beams with polarization along the vertical optical Z axis. The crystal was Y-cut (θ = π/2, |φ | = π/2) for noncritical (type 0) ee → e SHG; note that in [26] we mistakenly reported the crystal to be X-cut. This cut has Δk +500 mm −1 , resulting in a defocusing cascaded nonlinearity. Traditionally in type 0 cascading, periodic poling would be employed to reduce Δk and thereby increase the cascading strength, but we choose not to do this as it would leave the cascaded nonlinearity resonant and narrow-band [22,26,36]. Despite the large Δk, the cascaded nonlinearity is still strong because of the large d eff = d 33 = 25 pm/V. What matters is that the self-defocusing cascading nonlinearity is larger than the material self-focusing electronic Kerr nonlinearity so the pump effectively experiences a self-defocusing Kerr effect. An advantage of such a large Δk is actually that the SH conversion remains insignificant, typically on the order of a few percent. Note that the noncritical interaction has zero spatial walk-off (as opposed to the critical SHG scheme in [23,[27][28][29][30][31]).

Supercontinuum generation and MIR DW formation
With this setup significant spectral broadening occurred when gradually increasing the pump intensity, and Fig. 1 shows a supercontinuum under 1.4 μm and 0.8 TW/cm 2 pump condition. Similar spectra were observed at other pump wavelengths in the range λ 1 = 1.2 − 1.45 μm. Such an amount of spectral broadening is enabled by the ultra-broadband nature of the noncritical strongly phase-mismatched SHG process, which gives rise to soliton self-compression to few-cycle duration; indeed LN is predicted to support self-defocusing solitons in the pump regime we investigate [26]. After the soliton self-compression point a supercontinuum will form such as the one in Fig. 1, and it shows tremendous spectral extensions in the red side of the pump and goes far beyond the zero-dispersion wavelength λ ZD = 1.92 μm. As we will Normalized spectra recorded under various NIR pump wavelengths and using the maximum intensity available (see Table 1).
show below the MIR part of the supercontinuum spectrum is caused by the NIR soliton emitting a phase-matched DW in the anomalous GVD regime of LN, which turns out to be in the MIR (λ > λ ZD = 1.92 μm). The total bandwidth is around 1.5 octaves at −20 dB (spanning 163 THz, ranging from 3,175-8,600 cm −1 ). This exceeds the 1.1 octave that we previously measured with this setup [26] (however, note that in [26] the MIR part of the spectrum was not measured).
The MIR DWs appear as prominent peaks beyond the dip around 2.5 μm. We applied an LPF with the cut-on wavelength of 2.40 μm (Edmund Optics) to filter away the soliton and harvest the MIR DW. Figure 2(a) shows a typical evolution under various pump intensities; note that the LPF slope affects the spectrum up to 2.58 μm, shown with dashed lines. The MIR DW first appears at the pump intensity of 0.3 TW/cm 2 , and gradually builds up when the pump intensity further increases. At the maximum pump intensity the MIR bandwidth at −20 dB is 965 cm −1 , and covers the range from 3,000 cm −1 to 3,965 cm −1 .
The spatial quality of the filtered MIR pulses was also measured using an uncooled microbolometer camera (NEC, IRV-T0831). The inset of Fig. 2(a) shows a typical case, using Table 1. Properties of the MIR DWs from in Fig. 2(b), including input wavelength λ 1 , input energy W 1 and peak input intensity I 1 . The DW center wavelength λ DW is calculated as a weighted average over the filtered MIR spectrum. The MIR conversion efficiency η MIR was found by measuring the power of the filtered and unfiltered case. The MIR bandwidth Δν is shown as FWHM and at −20 dB. sian beam profile. It here is worth to mention that even under the maximum pump intensities, the output beam is stable and accompanied with relatively pure visible color from SHG. This confirms that the total nonlinearity is self-defocusing, enabling a filament-free supercontinuum.  Table 1). All these wavelengths lie in the normal dispersion regime for the LN crystal. This means that this range supports a self-defocusing soliton. A MIR peak centered at 3.00 μm was obtained with the 1.20 μm pump and a MIR peak centered at 2.83 μm was found from 1.45 μm pump. This shows a tunability over nearly 200 nm. This is the expected trend: when the pump wavelength decreases, the phase-matching wavelength of the MIR pulses increases. To see this, we need to look at the principle behind the DW formation, which is the well-known resonant phase-matching condition between a soliton and a linear (i.e. dispersive) wave. Specifically the DW phase-matching condition is k sol (ω) = k DW (ω), where k sol (ω) = k e (ω sol ) + (ω − ω sol )/v g,sol + q sol is the soliton wavenumber and k DW (ω) = k e (ω) is the DW wavenumber, which simply follows the material dispersion, in this case k e (ω) = n e (ω)ω/c. Here n e (ω) is the linear refractive index of the e-polarized pump. The dispersion-free nature of the soliton is reflected in its linear dependence in frequency, i.e. it is a wave-packet traveling with the group velocity v g,sol . There is also a nonlinear phase contribution to the soliton phase [37], which for cascaded SHG is q sol = n I 2,eff I sol ω sol /(2c) [23] where n I 2,eff = n I 2,casc + n I 2,Kerr,el is the effective nonlinear index, n I 2,casc = −2ω 1 d 2 eff /[ε 0 c 2 n 2 e (ω 1 )n e (ω 2 )Δk] the cascading nonlinear index and n I 2,Kerr,el = 3χ (3) /[4ε 0 cn 2 e (ω 1 )] is the Kerr electronic nonlinear index. For an effective defocusing nonlinearity (n I 2,eff < 0) this nonlinear contribution will shift the phase-matching condition slightly towards longer wavelengths. However, for most cases this shift is insignificant compared to the impact of shifting the NIR soliton wavelength; thus, the tunability of the MIR DW relies mainly on tuning the NIR soliton wavelength, and the phase-matching curve for LN with q sol = 0 can be found in Fig. 1(a) in [24].

MIR wavelength tunability
The wavelength tunability of the MIR DWs with pump wavelength was weaker than expected from theory. According to Fig. 1(a) in [24] a pump soliton in the 1.20 − 1.45 μm range should namely emit DWs with center wavelengths λ DW ∼ 3.0 − 5.0 μm, but instead we find λ DW ∼ 2.8−3.0 μm. Part of the explanation is to be found in the Raman-induced soliton self-frequency  [30,38]. The weak SH (containing less than 3% of the total energy) is not shown.
shift: LN has been found to have a significant Raman nonlinearity (see review in [39]), and the NIR soliton therefore will be significantly red-shifted compared to the input wavelength. This explains why DW radiation is found at shorter wavelengths than expected a priori from the input wavelength. However, the Raman soliton self-frequency shift alone cannot explain the results from Fig. 2.

Role of Raman effect in determining the MIR peak wavelengths
In order to understand this, we performed an extensive numerical study where we treated the total Kerr nonlinearity n I 2,Kerr,tot = n I 2,Kerr,el + n I 2,Kerr,R and the Raman fraction f R = n I 2,Kerr,R /n I 2,Kerr,tot as free parameters, but only varied within realistic limits of experimental measurements of n I 2,Kerr,tot and the Raman gain (see more in [39]). It was possible to locate quite broad parameter ranges where the dispersive wave emission was more or less fixed around the 2.8 − 3.0 μm regime. Figure 3 shows a characteristic example that showcases the typical dynamics well; f R = 0.35 and n I 2,Kerr,tot = 54 · 10 −20 m 2 /W was used. The large Raman fraction means that the Raman self-frequency shift will happen even before the soliton forms; as discussed in [40] the soliton spectral phase is affected by competing cascading and Raman effects, and when the Raman term is strong the spectrum will red-shift. We have illustrated this by indicating the soliton path, which was found by tracking the most intense part of the (τ, λ ) spectrogram. Thus the first appearance of the dispersive wave at the soliton formation point will be phase-matched to a soliton with a wavelength significantly longer than the pump. This explains why in Fig. 3 the dispersive wave initially (at the soliton self-compression point, black line in the cuts) is centered around 3.3 − 3.5 μm. This is significantly shorter than what one would expect with a soliton located at 1.4 μm, but it is consistent with the center wavelength of the main soliton. As the solitons are further Raman red-shifted the transfer of energy into dispersive waves becomes blue-shifted to below 3.0 μm. The solitons eventually experience the typical spectral recoil [8] as they approach the ZDW, and consequently the blue-shifting of the MIR dispersive waves is arrested. This is important because this seems to explain why despite the wide range of pump wavelengths and intensities that was used, we observed dispersive waves that were always centered around 2.8 − 3.0 μm. Instead a "traditional" scenario would predict Raman red-shifting only after soliton formation, so the initial dispersive wave peak position lies at longer wavelengths and only then is it blue-shifted as the soliton is gradually red-shifted.
Some characteristics of these numerical results are backed up by the experiment. First Fig.  4 show how the spectral broadening develops when increasing the intensity. Figure 4(a) shows the low-intensity NIR development, and significant broadening starts to occur between 50 and 90 GW/cm 2 , after which the new spectral shoulder clearly redshifts with increased intensity. This indicates that the Raman effect is indeed affecting the dynamics before soliton formation (no MIR DW radiation, the tell-tale sign of a soliton, was observed for these cases). Figure  4(b) shows the high-intensity development across the entire NIR and MIR; with these high intensities the MIR part of the spectrum started to appear. We note there are two DW peaks, since besides the main one just below 2.9 μm there is also a minor peak between 3.5 − 4.0 μm. The presence of this minor peak leads us to believe that it is an early-stage DW emitted at the self-compression point when the soliton first forms. The low-wavelength major peak would then be a DW emitted at a later stage. The reason why the low-wavelength DW is dominating in the spectrum can then be explained by the Raman red-shifting effect: at the initial self-compression point the soliton has only been red-shifted slightly, and has therefore a phase-matched DW far into the MIR. This DW will therefore have a weak coupling to the soliton and thereby a low radiation efficiency. As the soliton propagates further it relaxes and recompresses, where it again will emit a dispersive wave. During this stage the Raman effect will red-shift the soliton further, which leads to a phase-matching condition at a shorter MIR wavelength, and hence the coupling to the soliton is much stronger leading to a higher MIR DW radiation efficiency. As the soliton comes closer to the zero-dispersion wavelength it will be spectrally recoiled so its redshifting will be arrested, and therefore the blue-shifting of the DW emission will also arrest.
Going back to the low-intensity dynamics of Fig. 4(a), we note that the spectrum is substantially broadened and red-shifted but the MIR DW radiation was not observed for these cases. For sure the lowest intensities are not enough to form a soliton, but as the intensity is increased the soliton onset will occur at some point. It is here relevant to point out that once the soliton formation threshold is crossed, it is still required to see a significant spectral broadening in or- Fig. 4. Variation of the supercontinuum content at λ 1 = 1.4 μm while sweeping the input peak intensity. (a) The NIR development at low intensities (where no MIR spectral content was measurable); for comparison the spectrum for the maximum intensity is also shown. (b) The full spectra at high intensities (a 20 dB offset per curve is used for clarity). der to observe a MIR DW, i.e. a higher-order soliton must be excited. Once this happens the initial soliton self-compression point must occur within the relatively short crystal length we have chosen. Considering this, 200 GW/cm 2 might be enough for soliton self-compression to occur within 10 mm, and thereby also to form a DW. However, even if a soliton has formed already at this level, the coupling between the soliton and the DW is evidently not high enough for observing the MIR radiation. This would be in line the simulations of Fig. 3: at the soliton self-compression point the DW radiation is very weak, merely a bump on the soliton spectral tail. At 300 GW/cm 2 we do observe significant MIR radiation in form of DWs, and notice that the spectral shoulders of the soliton in the normal dispersion regime are located quite close to the zero-dispersion wavelength. This happens due to the Raman self-frequency shift. Usually a further increase in the intensity would lead to a stronger Raman red shift of the soliton spectrum, but the spectral recoil of the anomalous dispersion regime leads to a saturation of the red-shift: clearly the higher-intensity spectra do not present any change in the red-shift. This in turn leads to DWs that are emitted at roughly the same MIR wavelength, independent of the intensity. Therefore we conclude that the reason why the center wavelength of the major DW remains fixed while sweeping the intensity, as in Fig. 2(a), is because the Raman self-frequency shift leads to a significant red-shift of the spectrum, so that at the onset of soliton-formation in the 10 mm crystal the red-shift is already close to saturation caused by the spectral recoil of the anomalous dispersion regime.
As a consequence of the dynamics suggested by this explanation, we should also see that the MIR DW changes its peak position upon propagation. We performed experiments in shorter crystals (3 and 5 mm), and saw that the MIR spectra after the LPF initially (3 mm) is located beyond 3.0 μm and that only later (5 and 10 mm) the sub-3.0 μm spectrum appears.
The Raman self-frequency shift can also explain the main DW has a relatively limited tuning range from 2.8 − 3.0 μm. For low λ 1 the early-stage DW will be extremely weak because the phase-matching wavelength lies quite far away from the soliton. However, since the Raman effect will then shift the soliton we are able to observe a strong DW nonetheless, formed later in the crystal, but it is much more blue-shifted than what one would expect from the input wavelength as the soliton is now significantly red-shifted. Moreover, because the soliton red-shift will saturate, the low-wavelength input cases will not be so different from the high-wavelength input cases. That being said, we emphasize that there is a degree of tunability in the MIR pulses through the input wavelength, as Fig. 2(b) demonstrates, and we expect that by increasing the input wavelength further will extend the DW tunability to wavelengths shorter than 2.8 μm. It is also worth to note that LN is well-known to have a relatively strong Raman nonlinearity, so we expect that with other crystals with a less dominating Raman nonlinearity will show stronger tunability concerning the DW wavelength.